New insights into the normalization of the least mean fourth algorithm

The available normalization of the least mean fourth algorithm is investigated. It is shown that that normalization does not protect the algorithm from divergence when the input power of the adaptive filter increases. The reason of this drawback is that the normalization is done by dividing the weight vector update term by the squared norm of the regressor, while the update term is a fourth order polynomial in the regressor. The paper presents a normalized LMF algorithm that is based on dividing the weight vector update term by the fourth power of the norm of the regressor. This normalization protects the algorithm from divergence when the input power increases. An approximate stability step-size bound of the proposed algorithm is derived. The step-size bound depends on the weight initialization, while it does not depend on the input power of the adaptive filter for non-small signal-to-noise ratio. Simulation results support the analytical results of the paper.     

Finite precision arithmetic and the split Schur algorithms

The split Schur algorithms of P. Delsarte and Y. Genin (1987) represent methods of computing reflection coefficients that are computationally more efficient, in terms of multiplications, than the conventional Schur algorithm by a constant factor. The authors investigate the use of fixed-point binary arithmetic, with quantization due to rounding, in the implementation of the symmetric and antisymmetric split Schur algorithms. It is shown, through a combination of analysis and simulation, that the errors in the reflection coefficient estimates due to quantization are large when the input signal is either a narrowband high-pass signal or a narrowband low-pass signal     

一种全最小二乘算法及其在非线性滤波器中的应用

针对输入输出观测数据均含有噪声的滤波问题,提出了一种稳定的总体最小二乘自适应算法。该算法以系统的增广权向量的瑞利商(RQ)与对增广权向量的最后元素的约束的和作为总损失函数,利用梯度最陡下降原理导出权向量的自适应迭代算法,并将该算法应用于非线性Volterra滤波器。研究了算法的稳定性能,提出的算法不仅有良好的收敛性能,而且在权向量的自适应迭代时不需要标准化处理,使得算法的实施更为简单。仿真实验表明,无论在线性系统或非线性系统,本文算法的收敛性能,鲁棒抗噪性能和稳态收敛精度明显高于其它同类总体最小二乘算法。     

大规模MU-MIMO系统归一化预编码算法

针对大规模MU-MIMO系统中预编码技术性能不佳的问题,在不完善信道状态信息(CSI)的情况下,对迫零(ZF)和最大比发射(MRT)预编码技术提出了两种归一化算法：向量归一化与矩阵归一化。首先基站通过上行导频序列估计CSI,并在下行链路中用所提的算法对预编码矩阵进行归一化处理,然后将其与发送信号以及信道进行匹配。仿真结果表明,在高信噪比时,ZF预编码使用向量归一化算法实现了更好的系统性能;而在低信噪比时,MRT预编码使用矩阵归一化算法使系统性能得到了良好改善。     

Odd Symmetry of Weights Vector of Symmetrical Antenna Arrays with Linear Constraints

The paper provides proof of the odd symmetry of the vector of weight coefficients obtained on the basis of the least squares criterion in the linear antenna array with linear constraints and desired signal. Pairs of symmetrical elements of such vector are complex-conjugate to one another. For ensuring this property, the vector of constrained parameters (array pattern values in the directions of interest) must be real-valued, but need not be symmetrical. The odd symmetry of the antenna array vectors of input signals and weights makes it possible for such array to develop adaptive algorithms based on real-valued arithmetic. In this case, such algorithms have the number of arithmetic operations per iteration two or four times less as compared to the equivalent number of real-valued arithmetic operations of similar algorithms in the complex-valued arithmetic. The paper presents the results of comparative simulation of algorithms in the complex-valued and real-valued arithmetic. These results indicate that an adaptive algorithm using the real-valued arithmetic ensures (1.5–2)-fold shorter transient and deeper (by 2–3 dB) valleys in the steady state of the array pattern in the directions of sources of adaptively suppressed interferences as compared to the algorithm using the complex-valued arithmetic.     

A fast feature-based block matching algorithm using integralprojections

Block-by-block motion compensation algorithms are studied for video-conference/video-telephone television signals. A fast feature-based block matching algorithm using integral projections for the motion vector estimation is proposed. The proposed algorithm reduces the motion estimation computations by a factor of two by calculating the one-dimensional cost functions rather than the two-dimensional ones. Also, the low sensitivity of the proposed algorithm to the presence of additive noise is shown experimentally. Simulation results based on the original and noisy image sequences are presented     

共轭对称约束差分恒模波束形成方法

传统的恒模波束形成算法在期望信号的信噪比较低的情况下容易误收敛到干扰信号,导致波束形成算法失效。针对上述问题,本文提出了一种共轭对称约束差分恒模算法。该算法首先引入了一种新的差分恒模代价函数,然后根据均匀直线阵列导向矢量的特殊结构,推导出了共轭对称约束条件。最后通过对数据进行归一化处理,使得步长因子的选择独立于输入信号功率,降低了步长因子的选取难度,提高了算法的实用性。仿真实验表明,本文算法在低信噪比下仍然保持了良好的收敛性能,并且对阵列幅相误差和初始权重矢量的误差也具有较强的稳健性。      

A set of algorithms linking NLMS and block RLS algorithms

This paper describes a set of block processing algorithms which contains as extremal cases the normalized least mean squares (NLMS) and the block recursive least squares (BRLS) algorithms. All these algorithms use small block lengths, thus allowing easy implementation and small input-output delay. It is shown that these algorithms require a lower number of arithmetic operations than the classical least mean squares (LMS) algorithm, while converging much faster. A precise evaluation of the arithmetic complexity is provided, and the adaptive behavior of the algorithm is analyzed. Simulations illustrate that the tracking characteristics of the new algorithm are also improved compared to those of the NLMS algorithm. The conclusions of the theoretical analysis are checked by simulations, illustrating that, even in the case where noise is added to the reference signal, the proposed algorithm allows altogether a faster convergence and a lower residual error than the NLMS algorithm. Finally, a sample-by-sample version of this algorithm is outlined, which is the link between the NLMS and recursive least squares (RLS) algorithms     

鲁棒总体均方最小自适应滤波:算法与分析

本文研究了在输入输出观测数据均含有噪声的情况下如何有效地进行鲁棒自适应滤波的问题.以总体均方误差(TMSE)最小为准则,基于最速下降原理,通过对总体均方误差梯度进行修正,提出了一种鲁棒的总体均方最小自适应滤波算法.通过与已有算法的对比分析表明,该算法能够有效地降低权向量的每步调整量对噪声的敏感程度.仿真实验的结果进一步表明,该算法的鲁棒抗噪性能和稳态收敛精度明显地高于其它同类方法,而且可以使用较大的学习因子,在高噪声环境下仍然保持良好的收敛性.     

A new radix-2/8 FFT algorithm for length-q/spl times/2/sup m/ DFTs

In this paper, a new radix-2/8 fast Fourier transform (FFT) algorithm is proposed for computing the discrete Fourier transform of an arbitrary length N=q/spl times/2/sup m/, where q is an odd integer. It reduces substantially the operations such as data transfer, address generation, and twiddle factor evaluation or access to the lookup table, which contribute significantly to the execution time of FFT algorithms. It is shown that the arithmetic complexity (multiplications+additions) of the proposed algorithm is, in most cases, the same as that of the existing split-radix FFT algorithm. The basic idea behind the proposed algorithm is the use of a mixture of radix-2 and radix-8 index maps. The algorithm is expressed in a simple matrix form, thereby facilitating an easy implementation of the algorithm, and allowing for an extension to the multidimensional case. For the structural complexity, the important properties of the Cooley-Tukey approach such as the use of the butterfly scheme and in-place computation are preserved by the proposed algorithm.     

Analysis of the Error Signal of the LMS Algorithm

An analysis of the error signal of the Least-Mean-Square (LMS) algorithm is conducted from the robust control theory viewpoint. The difference equation that relates the input of the LMS algorithm and the error signal is presented. This equation is used to build the matrix ${bf S}$ that maps the input vector to the error vector. It is shown that ${bf S}$ has at least one singular value greater than 1. Therefore, the system may amplify noise at high frequencies. Nevertheless, the tap-weight vector may be chosen to prevent that noise amplification and improve the disturbance rejection performance of the LMS algorithm.      

A numerically stable fast Newton-type adaptive filter based on order recursive least squares algorithm

This paper presents a numerically stable fast Newton-type adaptive filter algorithm. Two problems are dealt with in the paper. First, we derive the proposed algorithm from an order-recursive least squares algorithm. The result of the proposed algorithm is equivalent to that of the fast Newton transversal filter (FNTF) algorithm. However, the derivation process is different. Instead of extending a covariance matrix of the input based on the min-max and the max-min criteria, the derivation shown in this paper is to solve an optimum extension problem of the gain vector based on the information of the Mth-order forward or backward predictor. The derivation provides an intuitive explanation of the FNTF algorithm, which may be easier to understand. Second, we present stability analysis of the proposed algorithm using a linear time-variant state-space method. We show that the proposed algorithm has a well-analyzable stability structure, which is indicated by a transition matrix. The eigenvalues of the ensemble average of the transition matrix are proved all to be asymptotically less than unity. This results in a much-improved numerical performance of the proposed algorithm compared with the combination of the stabilized fast recursive least squares (SFRLS) and the FNTF algorithms. Computer simulations implemented by using a finite-precision arithmetic have confirmed the validity of our analysis.     

Computation of an odd-length DCT from a real-valued DFT of the samelength

The discrete cosine transform (DCT) is often computed from a discrete Fourier transform (DFT) of twice or four times the DCT length. DCT algorithms based on identical-length DFT algorithms generally require additional arithmetic operations to shift the phase of the DCT coefficients. It is shown that a DCT of odd length can be computed by an identical-length DFT algorithm, by simply permuting the input and output sequences. Using this relation, odd-length DCT modules for a prime factor DCT are derived from corresponding DFT modules. The multiplicative complexity of the DCT is then derived in terms of DFT complexities     

A new split-radix FHT algorithm for length-q*2/sup m/DHTs

In this paper, a new split-radix fast Hartley transform (FHT) algorithm is proposed for computing the discrete Hartley transform (DHT) of an arbitrary length N=q*2/sup m/, where q is an odd integer. The basic idea behind the proposed FHT algorithm is that a mixture of radix-2 and radix-8 index maps is used in the decomposition of the DHT. This idea and the use of an efficient indexing process lead to a new decomposition different from that of the existing split-radix FHT algorithms, since the existing ones are all based on the use of a mixture of radix-2 and radix-4 index maps. The proposed algorithm reduces substantially the operations such as data transfer, address generation, and twiddle factor evaluation or access to the lookup table, which contribute significantly to the execution time of FHT algorithms. It is shown that the arithmetic complexity (multiplications+additions) of the proposed algorithm is, in almost all cases, the same as that of the existing split-radix FHT algorithm for length- q*2/sup m/ DHTs. Since the proposed algorithm is expressed in a simple matrix form, it facilitates an easy implementation of the algorithm, and allows for an extension to the multidimensional case.     

Fast RLS-type algorithm for unbiased equation-error adaptive IIR filtering based on approximate inverse-power iteration

In this paper, a fast approximate inverse-power (AIP) iteration is applied to compute recursively the total least squares (TLS) solution for unbiased equation-error adaptive infinite impulse response (IIR) filtering, which is established by approximating the well-known inverse-power iteration with Galerkin method. The AIP algorithm is based on an interesting modification of the inverse-power iteration in which the first entry of the parameter vector is constrained to the negative one. The distinctive feature of the proposed algorithm lies in its high computational efficiency, which is characterized by efficient computation of the fast gain vector (FGV), adaptation of the interesting modification of the inverse-power iteration, and rank-one update of the augmented correlation matrix. The shift structure of the input data vector is exploited to develop a fast algorithm for computing the gain vector. This FGV algorithm can be implemented at a numerical cost lower than the well-known fast Kalman algorithm. Since the first entry of the parameter vector has been fixed as the negative one and the weight vector is updated along the input data vector, a very efficient AIP algorithm is obtained by using the FGV algorithm. The proposed AIP algorithm is of computational complexity O(L) per iteration. Moreover, with no need to use the well-known matrix-inversion lemma, the AIP algorithm has another attractive feature of numerical stability. The proposed algorithm is shown to have global convergence. Simulation examples are included to demonstrate the effectiveness of the proposed AIP algorithm.     

Stochastic gradient based third-order Volterra system identification by using nonlinear Wiener adaptive algorithm

The nonlinear Wiener stochastic gradient adaptive algorithm for third-order Volterra system identification application with Gaussian input signals is presented. The complete self-orthogonalisation procedure is based on the delay-line structure of the nonlinear discrete Wiener model. The approach diagonalises the autocorrelation matrix of an adaptive filter input vector which dramatically reduces the eigenvalue spread and results in more rapid convergence speed. The relationship between the autocorrelation matrix and cross-correlation matrix of filter input vectors of both nonlinear Wiener and Volterra models is derived. The algorithm has a computational complexity of O(M/sup 3/) multiplications per sample input where M represents the length of memory for the system model, which is comparable to the existing algorithms. It is also worth noting that the proposed algorithm provides a general solution for the Volterra system identification application. Computer simulations are included to verify the theory.     

Optical processing using outer-product concepts

A row vector when left-multiplied by a column vector produces a two-dimensional rank-one matrix in an operation commonly called an outer product between the two vectors. The outer product operation can form the basis for a large variety of higher order algorithms in linear algebra, signal processing, and image processing. This operation can be best implemented in a processor having two-dimensional (2-D) parallelism and a global interaction among the elements of the input vectors. Since optics is endowed with exactly these features, an optical processor can perform the outer product operation in a natural fashion using orthogonally oriented one-dimensional (1-D) input devices such as acoustooptic cells. Algorithms that can be implemented optically using outer-product concepts include matrix multiplication, convolution/correlation, binary arithmetic operations for higher accuracy, matrix decompositions, and similarity transformations of images. Implementation is shown to be frequently tied to time-integrating detection techniques. These and other hardware issues in the implementation of some of these algorithms are discussed.     

On the convergence behavior of the LMS and the normalized LMSalgorithms

It is shown that the normalized least mean square (NLMS) algorithm is a potentially faster converging algorithm compared to the LMS algorithm where the design of the adaptive filter is based on the usually quite limited knowledge of its input signal statistics. A very simple model for the input signal vectors that greatly simplifies analysis of the convergence behavior of the LMS and NLMS algorithms is proposed. Using this model, answers can be obtained to questions for which no answers are currently available using other (perhaps more realistic) models. Examples are given to illustrate that even quantitatively, the answers obtained can be good approximations. It is emphasized that the convergence of the NLMS algorithm can be speeded up significantly by employing a time-varying step size. The optimal step-size sequence can be specified a priori for the case of a white input signal with arbitrary distribution     

A novel fingerprint image compression technique using wavelets packets and pyramid lattice vector quantization

A novel compression algorithm for fingerprint images is introduced. Using wavelet packets and lattice vector quantization , a new vector quantization scheme based on an accurate model for the distribution of the wavelet coefficients is presented. The model is based on the generalized Gaussian distribution. We also discuss a new method for determining the largest radius of the lattice used and its scaling factor , for both uniform and piecewise-uniform pyramidal lattices. The proposed algorithms aim at achieving the best rate-distortion function by adapting to the characteristics of the subimages. In the proposed optimization algorithm, no assumptions about the lattice parameters are made, and no training and multi-quantizing are required. We also show that the wedge region problem encountered with sharply distributed random sources is resolved in the proposed algorithm. The proposed algorithms adapt to variability in input images and to specified bit rates. Compared to other available image compression algorithms, the proposed algorithms result in higher quality reconstructed images for identical bit rates.